Animation stutters at beginning and end of loop
I should examine your .blend file to be sure, but I think that  the problem lies in the fact that your loop starts at 1 while the animation starts at 0. Instead of using 0, 8, 16, 24, 32....., use 1, 9, 17, 25, 33 ....
You'll end up with a 1-129 group of keyframes, go to keyframe 128, select all the rim and insert a new keyframe, then set the loop 1-128. If Keyframes 1 and 129 are identical it should loop perfectly.
Another way to loop an animation is via the Non Linear Editor (NLA).